number theory – Why does the proof of Theorem 3.10 in Apostol (1976) use generalized convolutions from previous chapter when there could be a simpler proof?

The proof of Theorem 3.10 in Introduction to Analytic Number Theory by Apostol goes like this:

Theorem 3.10 If $ h = f * g $, let
$$
H(x) = sum_{n le x} h(n),
F(x) = sum_{n le x} f(n), text{ and }
G(x) = sum_{n le x} g(n). $$

Then we have
$$
H(x)
= sum_{n le x} f(n) G left( frac{x}{n} right)
= sum_{n le x} g(n) F left( frac{x}{n} right). $$

PROOF. We make use of the associative law (Theorem 2.21) which relates the operations $ circ $ and $ * $. Let
$$
U(x) = begin{cases} 0 & text{ if } 0 < x < 1, \
1 & text{ if } x ge 1. end{cases}
$$

Then $ F = f circ U $, $ G = g circ U $, and we have
$$ f circ G = f circ (g circ U) = (f * g) circ U = H, $$
$$ g circ F = g circ (f circ U) = (g * f) circ U = H. $$
This completes the proof.

I am curious to understand why use generalized convolution and Dirichlet multiplication here at all.

Since section 3.5, the book has been relying on the following reordering of summation at various places,
$$
sum_{n le x} sum_{d | n} f(d)
= sum_{substack{q, d \ qd le x}} f(d)
= sum_{d le x} sum_{q le frac{x}{d}} f(q).
$$

Can we not arrive at a simpler proof with a similar reordering of summation like this:
begin{align*}
H(x)
& = sum_{n le x} h(n)
= sum_{n le x} (f * g)(n)
= sum_{n le x} sum_{d | n} f(d) gleft(frac{n}{d}right)
= sum_{substack{q, d \ qd le x}} f(d) gleft(qright) \
& = sum_{d le x} sum_{q le frac{x}{d}} f(d) g(q)
= sum_{d le x} f(d) sum_{q le frac{x}{d}} g(q)
= sum_{n le x} f(n) sum_{q le frac{x}{n}} g(q)
= sum_{n le x} f(n) Gleft(frac{x}{n}right).
end{align*}

I am trying to understand if a simpler proof using basic arithmetic and reordering of summations is correct or if I am missing something and there is a specific reason why Apostol went for a proof using generalized convolution and Dirichlet multiplication.